Is it possible to divide 1394 into 2 parts such that their HCF may be 34?

Answer

To determine if it's possible to divide 1394 into two parts such that their HCF (Highest Common Factor) is 34, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to divide the number 1394 into two parts, say \( a \) and \( b \), such that \( \text{HCF}(a, b) = 34 \).

2. **Expressing the Parts**: Since the HCF is 34, we can express \( a \) and \( b \) as:
\[ a = 34x \quad \text{and} \quad b = 34y \]
where \( x \) and \( y \) are co-prime integers (i.e., \( \text{HCF}(x, y) = 1 \)).

3. **Setting Up the Equation**: The sum of the two parts must equal 1394:
\[ a + b = 1394 \implies 34x + 34y = 1394 \]

4. **Factoring Out the Common Term**: We can factor out 34 from the left side:
\[ 34(x + y) = 1394 \]

5. **Dividing Both Sides by 34**: To isolate \( x + y \), we divide both sides by 34:
\[ x + y = \frac{1394}{34} \]

6. **Calculating the Right Side**: Now, we perform the division:
\[ \frac{1394}{34} = 41 \]
So, we have:
\[ x + y = 41 \]

7. **Finding Co-prime Pairs**: We need to find pairs of co-prime integers \( (x, y) \) such that their sum is 41.

8. **Checking Possible Pairs**: The pairs of integers that sum to 41 are:
- (1, 40)
- (2, 39)
- (3, 38)
- (4, 37)
- (5, 36)
- (6, 35)
- (7, 34)
- (8, 33)
- (9, 32)
- (10, 31)
- (11, 30)
- (12, 29)
- (13, 28)
- (14, 27)
- (15, 26)
- (16, 25)
- (17, 24)
- (18, 23)
- (19, 22)
- (20, 21)

9. **Identifying Co-prime Pairs**: From the above pairs, we check for co-primality. The pairs (1, 40), (2, 39), (3, 38), (4, 37), (5, 36), (6, 35), (8, 33), (9, 32), (10, 31), (11, 30), (12, 29), (13, 28), (14, 27), (15, 26), (16, 25), (17, 24), (18, 23), (19, 22), and (20, 21) all have at least one pair that is co-prime.

10. **Conclusion**: Since we can find co-prime pairs \( (x, y) \) such that \( x + y = 41 \), it is indeed possible to divide 1394 into two parts such that their HCF is 34.

### Final Answer:
Yes, it is possible to divide 1394 into two parts such that their HCF is 34.